A345301 a(n) = Sum_{p|n, p prime} p^pi(n/p).

0, 1, 1, 2, 1, 7, 1, 4, 9, 13, 1, 17, 1, 23, 52, 16, 1, 43, 1, 41, 130, 43, 1, 113, 125, 77, 81, 113, 1, 270, 1, 64, 364, 145, 968, 371, 1, 275, 898, 881, 1, 1328, 1, 377, 1354, 535, 1, 1241, 2401, 1137, 2476, 681, 1, 2699, 4456, 2913, 6922, 1053, 1, 10710, 1, 2079, 8962

Offset: 1

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

• nonn

Examples

			a(12) = Sum_{p|12} p^pi(12/p) = 2^pi(6) + 3^pi(4) = 2^3 + 3^2 = 17.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720.

Programs

Mathematica

Table[Sum[k^PrimePi[n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

PARI

A345301(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 1]^primepi(n/f[i, 1]))); \\ Antti Karttunen, Jan 22 2025

Python

from sympy import primefactors, primepi
def A345301(n): return sum(p**primepi(n//p) for p in primefactors(n)) # Chai Wah Wu, Jun 13 2021

Formula

a(p^k) = p^pi(p^(k-1)), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024